YITP-03-06 Marh 6, 2008 arxiv:hep-ph/030305v 6 Mar 2003 Event Shape/Energy Flow Correlations Carola F. Berger, Tibor Kús, and George Sterman C.N. Yang Institute for Theoretial Physis, Stony Brook University, SUNY Stony Brook, New York 794 3840, U.S.A. Abstrat We introdue a set of orrelations between energy flow and event shapes that are sensitive to the flow of olor at short distanes in jet events. These orrelations are formulated for a general set of event shapes, whih inludes jet broadening and thrust as speial ases. We illustrate the method for e + e dijet events, and alulate the orrelation at leading logarithm in the energy flow and at next-to-leading-logarithm in the event shape. Introdution The agreement of theoretial preditions with experiment for jet ross setions is often impressive. This is espeially so for inlusive jet ross setions at high p T, using fixed-order fatorized perturbation theory and parton distribution funtions []. A good deal is also known about the substruture of jets, through the theoretial and experimental study of multipliity distributions and fragmentation funtions [2], and of event shapes [3, 4, 5]. Event shape distributions [6, 7, 8] in partiular offer a bridge between the perturbative, short-distane and the nonperturbative, long-distane dynamis of QCD [9]. Energy flow [0] into angular regions between energeti jets gives information that is in some ways omplementary to what we learn from event shapes. In perturbation theory, the distribution
of partiles in the final state reflets interferene between radiation from different jets [2], and there is ample evidene for perturbative antenna patterns in interjet radiation at both e + e [] and hadron olliders [2, 3]. Energy flow between jets must also enode the mehanisms that neutralize olor in the hadronization proess, and the transition of QCD from weak to strong oupling. Knowledge of the interplay between energy and olor flows [4, 5] may help identify the underlying event in hadron ollisions [6], to distinguish QCD bremsstrahlung from signals of new physis. Nevertheless, the systemati omputation of energy flow into interjet regions has turned out to be subtle [7] for reasons that we will review below, and requires a areful onstrution of the lass of jet events. It is the purpose of this work to provide suh a onstrution, using event shapes as a tool. In this paper, we introdue orrelations between event shapes and energy flow, shape/flow orrelations, that are sensitive primarily to radiation from the highest-energy jets. So long as the observed energy is not too small, in a manner to be quantified below, we may ontrol logarithms of the ratio of energy flow to jet energy [5, 8]. The energy flow observables that we disuss below are distributions assoiated with radiation into a hosen interjet angular region, Ω. Within Ω we identify a kinemati quantity Q Ω εq, at.m. energy Q, with ε. Q Ω may be the sum of energies, transverse energies or related observables for the partiles emitted into Ω. Let us denote by Ω the omplement of Ω. We are interested in the distribution of Q Ω for events with a fixed number of jets in Ω. This set of events may be represented shematially as A + B Jets + X Ω + R Ω Q Ω ). ) Here X Ω stands for radiation into the regions between Ω and the jet axes, and R Ω for radiation into Ω. The subtlety assoiated with the omputation of energy flow onerns the origin of logarithms, and is illustrated by Fig.. Gluon in Fig. is an example of a primary gluon, emitted diretly from the hard partons near a jet axes. Phase spae integrals for primary emissions ontribute single logarithms per loop: /Q Ω )α n s ln n Q/Q Ω ) = /εq)α n s ln n /ε), n, and these logarithms exponentiate in a straightforward fashion [5]. At fixed Q Ω for Eq. ), however, there is another soure of potentially large logarithmi orretions in Q Ω. These are illustrated by gluon 2 in the figure, an example of seondary radiation in Ω, originating a parton emitted by one of the leading jets that define the event into intermediate region Ω. As observed by Dasgupta and Salam [7], emissions into Ω from suh seondary partons an also result in logarithmi orretions, of the form /Q Ω )α n s ln n Q Ω/Q Ω ), n 2, where Q Ω is the maximum energy emitted into Ω. These logarithms arise from strong ordering in the energies of the primary and seondary radiation beause real and virtual enhanements assoiated with seondary emissions do not anel eah other fully at fixed Q Ω. If the ross setion is fully inlusive outside of Ω, so that no restrition is plaed on the radiation into Ω, Q Ω an approah Q, and the seondary logarithms an beome as important as the primary logarithms. Suh a ross setion, in whih only radiation into a fixed portion of phase spae Ω) is speified, was termed non-global by Dasgupta and Salam, and the assoiated logarithms are also alled non-global [7, 9]. 2
In effet, a non-global definition of energy flow is not restritive enough to limit final states to a speifi set of jets, and non-global logarithms are produed by jets of intermediate energy, emitted in diretions between region Ω and the leading jets. Thus, interjet energy flow does not always originate diretly from the leading jets, in the absene of a systemati riterion for suppressing intermediate radiation. Correspondingly, non-global logarithms reflet olor flow at all sales, and do not exponentiate in a simple manner. Our aim in this paper is to formulate a set of observables for interjet radiation in whih non-global logarithms are replaed by alulable orretions, and whih reflet the flow of olor at short distanes. By restriting the sizes of event shapes, we will limit radiation in region Ω, while retaining the hosen jet struture. An important observation that we will employ below is that non-global logarithms are not produed by seondary emissions that are very lose to a jet diretion, beause a jet of parallelmoving partiles emits soft radiation oherently. By fixing the value of an event shape near the limit of narrow jets, we avoid final states with large energies in Ω away from the jet axes. At the same time, we will identify limits in whih non-global logarithms reemerge as leading orretions, and where the methods introdued to study nongobal effets in Refs. [7, 9] provide important insights. Å É Å ¾ É Å É Â Ø ½ ½ Å Â Ø ¾ Figure : Soures of global and non-global logarithms in dijet events. Configuration, a primary emission, is the soure of global logarithms. Configuration 2 an give non-global logarithms. To formalize these observations, we study below orrelated observables for e + e annihilation into two jets. In Eq. ) A and B denote positron and eletron.) In e + e annihilation dijet events, the underlying olor flow pattern is simple, whih enables us to onentrate on the energy flow within the event. We will introdue a lass of event shapes, fa) suitable for measuring energy flow into only part of phase spae, with a an adjustable parameter. To avoid large 3
non-global logarithmi orretions we weight events by exp[ ν f], with ν the Laplae transform onjugate variable. For the restrited set of events with narrow jets, energy flow is proportional to the lowest-order ross setion for gluon radiation into the seleted region. The resummed ross setion, however, remains sensitive to olor flow at short distanes through anomalous dimensions assoiated with oherent interjet soft emission. In a sense, our results show that an appropriate seletion of jet events automatially suppresses nonglobal logarithms, and onfirms the observation of oherene in interjet radiation [2, 2]. In the next setion, we introdue the event shapes that we will orrelate with energy flow, and desribe their relation to the thrust and jet broadening. Setion 3 ontains the details of the fatorization proedure that haraterizes the ross setion in the two-jet limit. This is followed in Se. 4 by a derivation of the resummation of logarithms of the event shape and energy flow, following the method introdued by Collins and Soper [20]. We then go on in Se. 5 to exhibit analyti results at leading logarithmi auray in Q Ω /Q and next-to-leading logarithm in the event shape. Setion 6 ontains representative numerial results. We onlude with a summary and a brief outlook on further appliations. 2 Shape/Flow Correlations 2. Weights and energy flow in dijet events In the notation of Eq. ), we will study an event shape distribution for the proess e + + e J p J ) + J 2 p J2 ) + X Ω f) + RΩ Q Ω ), 2) at.m. energy Q Q Ω Λ QCD. Two jets with momenta p J, =, 2 emit soft radiation only) at wide angles. Again, Ω is a region between the jets to be speified below, where the total energy or the transverse energy Q Ω of the soft radiation is measured, and Ω denotes the remaining phase spae see Fig. ). Radiation into Ω is onstrained by event shape f. We refer to ross setions at fixed values or transforms) of f and Q Ω as shape/flow orrelations. To impose the two-jet ondition on the states of Eq. 2) we hoose weights that suppress states with substantial radiation into Ω away from the jet axes. We now introdue a lass of event shapes f, related to the thrust, that enfore the two-jet ondition in a natural way. These event shapes interpolate between and extend the familiar thrust [4] and jet broadening [7, 8], through an adjustable parameter a. For eah state N that defines proess 2), we separate Ω into two regions, Ω, =, 2, ontaining jet axes, ˆn N). To be speifi, we let Ω and Ω 2 be two hemispheres that over the entire spae exept for their intersetions with region Ω. Region Ω is entered on ˆn, and Ω 2 is the opposite hemisphere. We will speify the method that determines the jet axes ˆn and ˆn 2 momentarily. To identify a meaningful jet, of ourse, the total energy within Ω should be a large fration of the available energy, of the order of Q/2 in dijet events. In e + e annihilation, if there is a well-ollimated jet in Ω with nearly half the total energy, there will automatially be one in Ω 2. 4
We are now ready to define the ontribution from partiles in region Ω to the a-dependent event shape, f Ω N, a) = k a s i, ω a i ˆn i ˆn ) a, 3) ˆn i Ω where a is any real number less than two, and where s = Q is the.m. energy. The sum is over those partiles of state N with diretion ˆn i that flow into Ω, and their transverse momenta k i, are measured relative to ˆn. The jet axis ˆn for jet is identified as that axis that minimizes the speifi thrust-related quantity f Ω N, a = 0). When Ω in Eq. 3) is extended to all of phase spae, the ase a = 0 is then essentially T, with T the thrust, while a = is related to the jet broadening. Any hoie a < 2 in 3) speifies an infrared safe event shape variable, beause the ontribution of any partile i to the event shape behaves as θi 2 a in the ollinear limit, θ i = os ˆn i ˆn ) 0. Negative values of a are learly permissible, and the limit a orresponds to the total ross setion. At the other limit, the fatorization and resummation tehniques that we disuss below will apply only to a <. For a >, ontributions to the event shape 3) from energeti partiles near the jet axis are generially larger than ontributions from soft, wide-angle radiation, or equal for a =. When this is the ase, the analysis that we present below must be modified, at least beyond the level of leading logarithm [8]. In summary, one ˆn is fixed, we have divided the phase spae into three regions: Region Ω, in whih we measure, for example, the energy flow, Region Ω, the entire hemisphere entered on ˆn, that is, around jet, exept its intersetion with Ω, Region Ω 2, the omplementary hemisphere, exept its intersetion with Ω. In these terms, we define the omplete event shape variable fn, a) by fn, a) = f Ω N, a) + f Ω2 N, a), 4) with f Ω, =, 2 given by 3) in terms of the axes ˆn of jet and ˆn 2 of jet 2. We will study the orrelations of this set of event shapes with the energy flow into Ω, denoted as fn) = s ˆn i Ω ω i. 5) The differential ross setion for suh dijet events at fixed values of f and f is now d σε, ε, s, a) dε d εdˆn = 2s MN) 2 2π) 4 δ 4 p I p N ) N δε fn)) δ ε fn, a)) δ 2 ˆn ˆnN)), 6) 5
where we sum over all final states N that ontribute to the weighted event, and where MN) denotes the orresponding amplitude for e + e N. The total momentum is p I, with p 2 I = s Q 2. Sine we are investigating energy flow in two-jet ross setions, we fix the onstants ε and ε to be both muh less than unity: 0 < ε, ε. 7) We refer to this as the elasti limit for the two jets. In the elasti limit, the dependene of the diretions of the jet axes on soft radiation is weak. We will return to this dependene below. Independent of soft radiation, we an always hoose our oordinate system suh that the transverse momentum of jet is zero, p J, = 0, 8) with p J in the x 3 diretion. In the limit ε, ε 0, and in the overall.m., p J and p J2 then approah light-like vetors in the plus and minus diretions: ) s p µ J 2, 0, 0 ) s 0 +, 2, 0. 9) p µ J 2 As usual, it is onvenient to work in light-one oordinates, p µ = p +, p, p ), whih we normalize as p ± = / 2)p 0 ± p 3 ). For small ε and ε, the ross setion 6) has orretions in ln/ε) and ln/ ε), whih we will organize in the following. 2.2 Weight funtions and jet shapes In Eq. 3), a is a parameter that allows us to study various event shapes within the same formalism; it helps to ontrol the approah to the two-jet limit. As noted above, a < 2 for infrared safety, although the fatorization that we will disuss below applies beyond leading logarithm only to > a >. A similar weight funtion with a non-integer power has been disussed in a related ontext for 2 > a > in [2]. To see how the parameter a affets the shape of the jets, let us reexpress the weight funtion for jet as f Ω N, a) = s ˆn i Ω ω i sin a θ i osθ i ) a, 0) where θ i is the angle of the momentum of final state partile i with respet to jet axis ˆn. As a 2 the weight vanishes only very slowly for θ i 0, and at fixed f Ω, the jet beomes very narrow. On the other hand, as a, the event shape vanishes more and more rapidly in the forward diretion, and the ross setion at fixed f Ω beomes more and more inlusive in the radiation into Ω. In this paper, as in Ref. [5], we seek to ontrol orretions in the single-logarithmi variable α s Q) ln/ε), with ε = Q Ω /Q. Suh a resummation is most relevant when ) α s Q) ln ε exp. ) ε) α s Q) 6
Å Å Ð ¾ ¾Æ ½ Figure 2: A kinemati onfiguration that gives rise to the non-global logarithms. A soft gluon with momentum k is radiated into the region Ω, and an energeti gluon with momentum l is radiated into Ω. Four-vetors β and β 2, define the diretions of jet and jet 2, respetively. Let us ompare these logarithms to non-global effets in shape/flow orrelations. At ν = 0 and for a, the ross setion beomes inlusive outside Ω. As we show below, the non-global logarithms disussed in Refs. [5, 7] appear in shape/flow orrelations as logarithms of the form α s Q) ln/εν)), with ν the moment variable onjugate to the event shape. To treat these logarithms as subleading for small ε and relatively) large ν, we require that ) α s Q) ln < ε > ) εν ν exp. 2) α s Q) For large ν, there is a substantial range of ε in whih both ) and 2) an hold. When ν is large, moments of the orrelation are dominated preisely by events with strongly two-jet energy flows, whih is the natural set of events in whih to study the influene of olor flow on interjet radiation. The peak of the thrust ross setion is at T) of order one-tenth at LEP energies, orresponding to ν of order ten, so the requirement of large ν is not overly restritive.) In the next subsetion, we show how the logarithms of εν) emerge in a low order example. This analysis also assumes that a is not large in absolute value. The event shape at fixed angle dereases exponentially with a, and we shall see that higher-order orretions an be proportional to a. We always treat ln ν as muh larger than a. 2.3 Low order example In this setion, we hek the general ideas developed above with the onrete example of a twoloop ross setion for the proess 2). This is the lowest order in whih a non-global logarithm ours, as observed in [7]. We normalize this ross setion to the Born ross setion for inlusive dijet prodution. A similar analysis for the same geometry has been arried out in [7] and [22]. The kinemati onfiguration we onsider is shown in Fig. 2. Two fast partons, of veloities β and β 2, are treated in eikonal approximation. In addition, gluons are emitted into the final state. 7
½ ½ ½ Ð Ð Ð Ð Ð ¾ ¾ ¾ µ µ µ ½ ½ ½ Ð Ð Ð Ð ¾ ¾ ¾ µ µ µ Figure 3: The relevant two-loop ut diagrams orresponding to the emission of two real gluons in the final state ontributing to the eikonal ross setion. The dashed line represents the final state, with ontributions to the amplitude to the left, and to the omplex onjugate amplitude to the right. A soft gluon with momentum k is radiated into region Ω and an energeti gluon with momentum l is emitted into the region Ω. We onsider the ross setion at fixed energy, ω k ε s. As indiated above, non-global logarithms arise from strong ordering of the energies of the gluons, whih we hoose as ω l ω k. In this region, the gluon l plays the role of a primary emission, while k is a seondary emission. For our alulation, we take the angular region Ω to be a slie or ring in polar angle of width 2δ, or equivalently, pseudo) rapidity interval η, η), with η = 2η = ln ) + sin δ, 3) sin δ The lowest-order diagrams for this proess are those shown in Fig. 3, inluding distinguishable diagrams in whih the momenta k and l are interhanged. The diagrams of Fig. 3 give rise to olor strutures CF 2 and C FC A, but terms proportional to CF 2 may be assoiated with a fatorized ontribution to the ross setion, in whih the gluon k is emitted oherently by the ombinations of the gluon l and the eikonals. To generate the C F C A part, on the other hand, gluon k must resolve gluon l from the eikonal lines, giving a result that depends on the angles between l and the eikonal diretions. 8
The omputation of the diagrams is outlined in Appendix A; here we quote the results. We adopt the notation l osθ l, s l sin θ l, with θ l the angle of momentum l measured relative to β, and similarly for k. We take, as indiated above, a Laplae transform with respet to the shape variable, and identify the logarithm in the onjugate variable ν. We find that the logarithmi C F C A -dependene of Fig. 3 may be written as a dimensionless eikonal ross setion in terms of one energy and two polar angular integrals as dσ eik d ε αs = C F C A π [ ) 2 ε k + l + k sinδ sinδ d k sinδ + l + k s dω l d l ε e ν ω l l ) a sa l /Q s ω l ) ]. 4) s 2 k + l In this form, the absene of ollinear singularities in the C F C A term at osθ l = + is manifest, independent of ν. Collinear singularities in the l integral ompletely fatorize from the k integral, and are proportional to CF 2. The logarithmi dependene on ε for ν > is readily found to be dσ eik d ε = C FC A ) αs 2 ) π ε ln C η), 5) εν where C η) is a finite funtion of the angle δ, given expliitly in Appendix A. We an ontrast this result to what happens when ν = 0, that is, for an inlusive, non-global ross setion. In this ase, realling that ε = Q Ω /Q, we find in plae of Eq. 5) the non-global logarithm dσ eik d ε = C FC A ) ) αs 2 Q π ε ln C η). 6) Q Ω As antiipated, the effet of the transform is to replae the non-global logarithm in Q/Q Ω, by a logarithm of /εν). We are now ready to generalize this result, starting from the fatorization properties of the ross setion near the two-jet limit. 3 Fatorization of the Cross Setion 3. Leading regions near the two-jet limit In order to resum logarithms of ε and ε or equivalently ν, the Laplae onjugate of ε) we have first to identify their origin in momentum spae when ε, ε 0. Following the proedure and terminology of [23], we identify leading regions in the momentum integrals of ut diagrams, whih an give rise to logarithmi enhanements of the ross setion assoiated with lines approahing the mass shell. Within these regions, the lines of a ut diagram fall into the following subdiagrams: A hard-sattering, or short-distane subdiagram H, where all omponents of line momenta are far off-shell, by order Q. 9
Jet subdiagrams, J and J 2, where energies are fixed and momenta are ollinear to the outgoing primary partons and the jet diretions that emerge from the hard sattering. For ε = ε = 0, the sum of all energies in eah jet is one-half the total energy.) To haraterize the momenta of the lines within the jets, we introdue a saling variable, λ. Within jet, momenta l sale as l + Q, l λq, l λ /2 Q). A soft subdiagram, S onneting the jet funtions J and J 2, in whih the omponents of momenta k are small ompared to Q in all omponents, saling as k ± λq, k λq). An arbitrary final state N is the union of substates assoiated with these subdiagrams: N = N s N J N J2. 7) As a result, the event shape f an also be written as a sum of ontributions from the soft and jet subdiagrams: fn, a) = f N N s, a) + f N Ω N J, a) + f N Ω N J2, a). 8) The supersript N reminds us that the ontributions of final-state partiles assoiated with the soft and jet funtions depend impliitly on the full final state, through the determination of the jet axes, as disussed in Se. 2. In ontrast, the energy flow weight, fn), depends only on partiles emitted at wide angles, and is hene insensitive to ollinear radiation: fn) = fn s ). 9) When we sum over all diagrams that have a fixed final state, the ontributions from these leading regions may be fatorized into a set of funtions, eah of whih orresponds to one of the generi hard, soft and jet subdiagrams. The arguments for this fatorization at leading power have been disussed extensively [20, 24, 25]. The ross setion beomes a onvolution in ε, with the sums over states linked by the delta funtion whih fixes ˆn, and by momentum onservation, d σε, ε, s, a) dε d εdˆn = dσ 0 dˆn Hs, ˆn ) 2 = N s,n J d ε s SN s ) δε fn s )) δ ε s f N N s, a)) d ε J J N J ) δ ε J f N Ω N J, a)) 2π) 4 δ 4 p I pn J2 ) pn J ) pn s )) δ 2 ˆn ˆnN)) δ ε ε J ε J2 ε s ) = dσ 0 dˆn δε) δ ε) + Oα s ). 20) Here dσ 0 /dˆn is the Born ross setion for the prodution of a single partile quark or antiquark) in diretion ˆn, while the short-distane funtion Hs, ˆn ) = + Oα s ), whih desribes orretions to the hard sattering, is an expansion in α s with finite oeffiients. The funtions 0
J N J ), SN s ) desribe the internal dynamis of the jets and wide-angle soft radiation, respetively. We will speify these funtions below. We have suppressed their dependene on a fatorization sale. Radiation at wide angles from the jets will be well-desribed by our soft funtions SN s ), while we will onstrut the jet funtions J N J ) to be independent of ε, as in Eq. 20). So far, we have speified our sums over states in Eq. 20) only when all lines in N s are soft, and all lines in N J have momenta that are ollinear, or nearly ollinear to p J. As ε and ε vanish, these are the only final-state momenta that are kinematially possible. Were we to restrit ourselves to these onfigurations only, however, it would not be straightforward to make the individual sums over N s and N J infrared safe. Thus, it is neessary to inlude soft partons in N s that are emitted near the jet diretions, and soft partons in the N J at wide angles. We will show below how to define the funtions J N J ), SN s ) so that they generate fatoring, infrared safe funtions that avoid double ounting. We know on the basis of the arguments of Refs. [20, 24, 25] that orretions to the fatorization of soft from jet funtions are suppressed by powers of the weight funtions ε and/or ε. 3.2 The fatorization in onvolution form Although formally fatorized, the jet and soft funtions in Eq. 20) are still linked in a potentially ompliated way through their dependene on the jet axes. Our strategy is to simplify this omplex dependene to a simple onvolution in ontributions to ε, aurate to leading power in ε and ε. First, we note that the ross setion of Eq. 20) is singular for vanishing ε and ε, but is a smooth funtion of s and ˆn. We may therefore make any approximation that hanges s and/or ˆn by an amount that vanishes as a power of ε and ε in the leading regions. Correspondingly, the amplitudes for jet are singular in ε J, but depend smoothly on the jet energy and diretion, while the soft funtion is singular in both ε and ε s, but depends smoothly on the jet diretions. As a result, at fixed values of ε and ε we may approximate the jet diretions and energies by their values at ε = ε = 0 in the soft and jet funtions. Finally, we may make any approximation that affets the value of ε and/or ε J by amounts that vanish faster than linearly for ε 0. It is at this stage that we will require that a <. With these observations in mind, we enumerate the replaements and approximations by whih we redue Eq. 20), while retaining leading-power auray.. To simplify the definitions of the jets in Eq. 20), we make the replaements f N Ω N J, a) f N J, a) with f N J, a) s all ˆn i N J k a i, ω a i ˆn i ˆn ) a. 2) The jet weight funtion f N J, a) now depends only on partiles assoiated with N J. The ontribution to f N J, a) from partiles within region Ω, is exatly the same here as in the weight 3), but we now inlude partiles in all other diretions. In this way, the independent
sums over final states of the jet amplitudes will be naturally infrared safe. The value of f N J, a) differs from the value of fn Ω N J, a), however, due to radiation outside Ω, as indiated by the new subsript. This radiation is hene at wide angles to the jet axis. In the elasti limit 7), it is also onstrained to be soft. Double ounting in ontributions to the total event shape, fn, a), will be avoided by an appropriate definition of the soft funtion below. The sums over states are still not yet fully independent, however, beause the jet diretions ˆn still depend on the full final state N. 2. Next, we turn our attention to the ondition that fixes the jet diretion ˆn. Up to orretions in the orientation of ˆn that vanish as powers of ε and ε, we may neglet the dependene of ˆn on N s and N J2 : δˆn ˆnN)) δˆn ˆnN J )). 22) In Appendix B, we show that this replaement also leaves the value of ε unhanged, up to orretions that vanish as ε 2 a. Thus, for a <, 22) is aeptable to leading power. For a <, we an therefore identify the diretion of jet with ˆn. These approximations simplify Eq. 20) by eliminating the impliit dependene of the jet and soft weights on the full final state. We may now treat ˆn as an independent vetor. 3. In the leading regions, partiles that make up eah final-state jet are assoiated with states N J, while N s onsists of soft partiles only. In the momentum onservation delta funtion, we an neglet the four-momenta of lines in N s, whose energies all vanish as ε, ε 0: δ 4 p I pn J2 ) pn J ) pn s )) δ 4 p I p J2 p J ). 23) 4. Beause the ross setion is a smooth funtion of the jet energies and diretions, we may also neglet the masses of the jets within the momentum onservation delta funtion, as in Eq. 9). In this approximation, we derive in the.m., δ 4 p I p J2 p J ) δ s ωn J ) ωn J2 )) δ p J p J2 ) 2 ) s s δ 2 ωn J ) δ p J 2 δ2 ˆn + ˆn 2 ) ) s 2 ωn J 2 ) δ 2 ˆn + ˆn 2 ). 24) Our jets are now bak-to-bak: ˆn 2 ˆn. 25) Implementing these replaements and approximations for a <, we rewrite the ross setion Eq. 20) as d σε, ε, s, a) dε d εdˆn = dσ 0 dˆn Hs, ˆn, µ) 2 = d ε s Sε, εs, a, µ) d ε J J ε J, a, µ) δ ε ε J ε J2 ε s ), 26) 2
with as above) H = + Oα s ). Referring to the notation of Eqs. 20) and 2), the funtions S and J are: Sε, ε s, a, µ) = Ns SN s, µ) δε fn s )) δ ε s fn s, a)) 27) J ε J, a, µ) = 2 s 2π)6 NJ J N J, µ) δ ε J f N J, a)) δ ) s 2 ωn J ) δ 2 ˆn ± ˆnN J )), with the plus sign in the angular delta funtion for jet 2, and the minus for jet. The weight funtions for the jets are given by Eq. 2) and indue dependene on the parameter a. We have introdued the fatorization sale µ, whih we set equal to the renormalization sale. We note that we must onstrut the soft funtions SN s, µ) to anel the ontributions of final-state partiles from eah of the J N J, µ) to the weight ε, as well as the ontributions of the jet funtions to ε from soft radiation outside their respetive regions Ω. Similarly, the jet amplitudes must be onstruted to inlude ollinear enhanements only in their respetive jet diretions. Expliit onstrutions that satisfy these requirements will be speified in the following subsetions. To disentangle the onvolution in 26), we take Laplae moments with respet to ε: dσε, ν, s, a) dε dˆn = 0 ν ε d σε, ε, a) d εe dε d εdˆn = dσ 0 dˆn Hs, ˆn, µ) Sε, ν, a, µ) 2 = 28) J ν, a, µ). 29) Here and below unbarred quantities are the transforms in ε, and barred quantities denote untransformed funtions. Sε, ν, a, µ) = d ε s e ν εs Sε, ε s, a, µ), 30) 0 and similarly for the jet funtions. In the following subsetions, we give expliit onstrutions for the funtions partiipating in the fatorization formula 26), whih satisfy the requirement of infrared safety, and avoid double ounting. An illustration of the ross setion fatorized into these funtions is shown in Fig. 4. As disussed above, non-global logarithms will emerge when εν beomes small enough. 3.3 The short-distane funtion The power ounting desribed in [23] shows that in Feynman gauge the subdiagrams of Fig. 4 that ontribute to H in Eq. 26) at leading power in ε and ε are onneted to eah of the two jet subdiagrams by a single on-shell quark line, along with a possible set of on-shell, ollinear gluon lines that arry salar polarizations. The hard subdiagram is not onneted diretly to the soft subdiagram in any leading region. 3
 ½ É Ô ½ À Ë À Ô ¾ ½ ¾ ½ ¾  µ ½  µ ¾  ¾ Figure 4: Fatorized ross setion 26) after the appliation of Ward identities. The vertial line denotes the final state ut. The ouplings of the salar-polarized gluons that onnet the jets with short-distane subdiagrams may be simplified with the help of Ward identities see, e.g. [25]). At eah order of perturbation theory, the oupling of salar-polarized gluons from either jet to the short-distane funtion is equivalent to their oupling to a path-ordered exponential of the gauge field, oriented in any diretion that is not ollinear to the jet. Corretions are infrared safe, and an be absorbed into the short-distane funtion. Let hp J, ˆn, A) represent the set of all short-distane ontributions to diagrams that ouple any number of salar-polarized gluons to the jets, in the amplitude for the prodution of any final state. The argument A stands for the fields that reate the salar-polarized gluons linking the short-distane funtion to the jets. On a diagram-bydiagram basis, h depends on the momentum of eah of the salar-polarized gluons. After the sum over all diagrams, however, we an make the replaement: hp J, ˆn, A q, q) ) Φ q) ξ 2 0, ; 0) h 2 p J, ˆn, ξ ) Φ q) ξ 0, ; 0), 3) where h 2 is a short-distane funtion that depends only on the total momenta p J and p J2. It also depends on vetors ξ that haraterize the path-ordered exponentials Φ0, ; 0): Φ f) ξ 0, ; 0) = Pe ig 0 dλ ξ Af) λξ ), 32) where the supersript f) indiates that the vetor potential takes values in representation f, in our ase the representation of a quark or antiquark. These operators will be assoiated with gauge-invariant definitions of the jet funtions below. To avoid spurious ollinear singularities, we hoose the vetors ξ, =, 2, off the light one. In the full ross setion 29) the ξ -dependene anels, of ourse. 4
The dimensionless short-distane funtion H = h 2 2 in Eq. 26) depends on s and p J ξ, but not on any variable that vanishes with ε and ε: s Hp J, ξ, ˆn, µ) = H µ, p J ˆξ ) µ, ˆn, α s µ), 33) where ˆξ ξ / ξ 2. 34) Here we have observed that eah diagram is independent of the overall sale of the eikonal vetor ξ µ. 3.4 The jet funtions The jet funtions and the soft funtions in Eq. 26) an be defined in terms of speifi matrix elements, whih absorb the relevant ontributions to leading regions in the ross setion, and whih are infrared safe. Their perturbative expansions speify the funtions S and J of Eq. 28). We begin with our definition of the jet funtions. The jet funtions, whih absorb enhanements ollinear to the two outgoing partiles produed in the primary hard sattering, an be defined in terms of matrix elements in a manner reminisent of parton distribution or deay funtions [26]. To be speifi, we onsider the quark jet funtion: J µ ε J, a, µ) = 2 s 2π) 6 N C N J Tr [ γ µ 0 Φ q) ξ 0, ; 0)q0) NJ NJ q0)φ q) ξ 0, ; 0) 0 ] δ ε J f N J, a)) δ ) s 2 ωn J ) δ 2 ˆn ˆnN J )), 35) where N C is the number of olors, and where ˆn denotes the diretion of the momentum of jet, Eq. 28), with ˆn 2 = ˆn. q is the quark field, Φ q) ξ 0, ; 0) a path-ordered exponential in the notation of 32), and the trae is taken over olor and Dira indies. We have hosen the normalization so that the jet funtions J µ in 35) are dimensionless and begin at lowest order with J µ 0) ε J, a, µ) = β µ δ ε J ), 36) with β µ the lightlike veloities orresponding to the jet momenta in Eq. 9): β µ = δ µ+, β µ 2 = δ µ. 37) The salar jet funtions of Eq. 28) are now obtained by projeting out the omponent of J µ in the jet diretion: J ε J, a, µ) = β J ε J, a, µ) = δ ε J ) + Oα s ), 38) where β = β 2, β2 = β are the lightlike vetors in the diretions opposite to β and β 2, respetively. By onstrution, the J are linear in β. 5
To resum the jet funtions in the variables ε J, it is onvenient to reexpress the weight funtions 2) in ombinations of light-one momentum omponents that are invariant under boosts in the x 3 diretion, f N J, a) = f 2 N J2, a) = s a/2 s a/2 ˆn i N J k a i, ˆn i N J2 k a i, 2p + J k i 2p J2 k + i ) a, ) a. 39) Here we have used the relation s/2 = ω J, valid for both jets in the.m. At the same time, we make the identifiation, ) s s δ 2 ωn J ) δ 2 ˆn ˆnN J )) = 4 δ3 p J pn J )), 40) whih again holds in the.m. frame. The spatial omponents of eah p J are thus fixed. Given that we are at small ε J, the jet funtions may be thought of as funtions of the light-like jet momenta p µ J of Eq. 9) and of ε J. Beause the vetor jet funtion is onstruted to be dimensionless, J µ in Eq. 35) is proportional to β rather than p J. Otherwise, it is free of expliit β -dependene. The jet funtions an now be written in terms of boost-invariant arguments, homogeneous of degree zero in ξ : J ε J, a, µ) = β µ [ β µ J ) + 2 ξµ β ξ ξ 2 J2) p J ˆξ ) a s s µ, ε J µ 2p J ˆξ, a, α s µ) p J ˆξ ) a ] s s µ, ε J µ 2p J ˆξ, a, α s µ), 4) where J ) and J 2) are independent funtions, and where we have suppressed possible dependene on ˆξ,. For jet, the weight ε J is fixed by δ ε J f N J, a)), where on the right-hand side of the expression for the weight 39), the sum over eah partile s momentum involves the overall fator 2p ± J / s) a. After integration over final states at fixed ε J, the jet an thus depend on the vetor p µ J. At the same time, it is easy to see from the definition of the weight that p µ J an only appear in the ombination / ε J s) / a) 2p µ J / s). This vetor an ombine with ξ to form an invariant, and all ξ -dependene omes about in this way. Expression 4) an be further simplified by noting that Choosing ξ, = 0, we find a single ombination, 2 β ξ β ξ = ξ 2 + ξ2,. 42) J ε J, a, µ) = J pj ˆξ ) s µ, ε J µ ζ ) a, a, α s µ), 43) 6
where, in the notation of Eq. 4), J = ) J + 2) J, and we have defined ζ s 2p J ˆξ. 44) In these terms, the Laplae moments of the jet funtion inherit dependene on the moment variable ν through J ν, a, µ) = 0 d ε J e ν ε J J ε J, a, µ) J pj ˆξ µ, ) s µν ζ ) a, a, α s µ), 45) where the unbarred and barred quantities denote transformed and untransformed funtions, respetively. We have onstruted the jet funtions to be independent of ε, sine the radiation into Ω is at wide angles from the jet axes and an therefore be ompletely fatored from the ollinear radiation. This radiation at wide angles is ontained in the soft funtion, whih will be defined below in a manner that avoids double ounting in the ross setion. 3.5 The soft funtion Given the definitions for the jet funtions in the previous subsetion, and the fatorization 26), we may in priniple alulate the soft funtion S order by order in perturbation theory. We an derive a more expliit definition of the soft funtion, however, by relating it to an eikonal analog of Eq. 26). As reviewed in Refs. [5, 25], soft radiation at wide angles from the jets deouples from the ollinear lines within the jet. As a result, to ompute amplitudes for wide-angle radiation, the jets may be replaed by nonabelian phases, or Wilson lines. We therefore onstrut a dimensionless quantity, σ eik), in whih gluons are radiated by path-ordered exponentials Φ, whih mimi the olor flow of outgoing quarks, β, 0; x) = Pe ig dλβ 0 A f) λβ +x), 46) Φ f) with β a light-like veloity in either of the jet diretions. For the two-jet ross setion at measured ε and ε eik, we define σ eik) ε, ε eik, a, µ) N C N eik Φ q) 0 β 2, 0; 0)Φ q) β, 0; 0) N eik Φ q) N eik β, 0; 0)Φ q) β 2, 0; 0) 0 δ ε fn eik )) δ ε eik fn eik, a) ) = δε) δ ε eik ) + Oα s ). 47) The sum is over all final states N eik in the eikonal ross setion. The renormalization sale in this ross setion, whih will also serve as a fatorization sale, is denoted µ. Here the event 7
shape funtion ε eik is defined by fn eik, a) as in Eqs. 3) and 4), distinguishing between the hemispheres around the jets. As usual, N C is the number of olors, and a trae over olor is understood. The eikonal ross setion 47) models the soft radiation away from the jets, inluding the radiation into Ω, aurately. It also ontains enhanements for onfigurations ollinear to the jets, whih, however, are already taken into aount by the partoni jet funtions in 26). Indeed, 47) does not reprodue the partoni ross setion aurately for ollinear radiation. It is also easy to verify at lowest order that even at fixed ε eik the eikonal ross setion 47) is ultraviolet divergent in dimensional regularization, unless we also impose a utoff on the energy of real gluon emission ollinear to β or β 2. The onstrution of the soft funtion S from σ eik) is nevertheless possible beause the eikonal ross setion 47) fatorizes in the same manner as the ross setion itself, into eikonal jet funtions and a soft funtion. The essential point [4] is that the soft funtion in the fatorized eikonal ross setion is the same as in the original ross setion 26). The eikonal jets organize all ollinear enhanements in 47), inluding the spurious ultraviolet divergenes. These eikonal jet funtions are defined analogously to their partoni ounterparts, Eq. 35), but now with ordered exponentials replaing the quark fields, J eik) ε, a, µ) Φ f 0 ) ξ 0, ; 0)Φ f) β, 0; 0) N eik) N C N eik) N eik) Φ f) β, 0; 0)Φ f) ξ 0, ; 0) 0 δ ε f N eik), a) ) = δ ε ) + Oα s ), 48) where f is a quark or antiquark, and where the trae over olor is understood. The weight funtions are given as above, by Eq. 2), with the sum over partiles in all diretions. In terms of the eikonal jets, the eikonal ross setion 47) fatorizes as σ eik) ε, ε eik, a, µ) d ε s S ε, εs, a, µ) 2 = d ε Jeik) ε, a, µ) δ ε eik ε s ε ε 2 ), 49) where we pik the fatorization sale equal to the renormalization sale µ. As for the full ross setion, the onvolution in 49) is simplified by a Laplae transformation 45) with respet to ε eik, whih allows us to solve for the soft funtion as S ε, ν, a, µ) = σeik) ε, ν, a, µ) = δε) + Oα 2 s ). 50) J eik) ν, a, µ) = In this ratio, ollinear logarithms in ν and the unphysial ultraviolet divergenes and their assoiated utoff dependene anel between the eikonal ross setion and the eikonal jets, leaving a soft funtion that is entirely free of ollinear enhanements. The soft funtion retains ν-dependene through soft emission, whih is also restrited by the weight funtion ε. In addition, beause soft radiation within the eikonal jets an be fatored from its ollinear radiation, just as in the 8
partoni jets, all logarithms in ν assoiated with wide-angle radiation are idential between the partoni and eikonal jets, and fator from logarithmi orretions assoiated with ollinear radiation in both ases. As a result, the inverse eikonal jet funtions anel ontributions from the wide-angle soft radiation of the partoni jets in the transformed ross setion 29). Given the definition of the energy flow weight funtion f, Eq. 5), the soft funtion is not boost invariant. In addition, beause it is free of ollinear logs, it an have at most a single logarithm per loop. Its dependene on ε is therefore only through ratios of the dimensional quantities ε s with the renormalization fatorization) sale. As in the ase of the partoni jets, Eq. 45), we need to identify the variable through whih ν appears in the soft funtion. We note that dependene on the veloity vetors β and the fatorization vetors ξ must be sale invariant in eah, sine they arise only from eikonal lines and verties. The eikonal jet funtions annot depend expliitly on the sale-less, lightlike eikonal veloities β, and σ eik) is independent of the ξ. Dependene on the fatorization vetors ξ enters only through the weight funtions, 39) for the eikonal jets, in a manner analogous to the ase of the partoni jets. This results in a dependene on ζ ) a, as above, with ζ defined in Eq. 44). In summary, we may haraterize the arguments of the soft funtion in transform spae as 4 Resummation S ε, ν, a, µ) = S ε s µ, εν, s µν ζ ) a, a, α s µ) ). 5) We may summarize the results of the previous setion by rewriting the transform of the fatorized ross setion 29) in terms of the hard, jet and soft funtions identified above, whih depend on the kinemati variables and the moment ν aording to Eqs. 33), 45) and 5) respetively, dσ ε, ν, s, a) dε dˆn = dσ 0 dˆn H s µ, p J ˆξ S ) 2 µ, ˆn pj, α s µ) J ˆξ = µ, ) s µν ζ ) a, a, α s µ) ε s µ, εν, s µν ζ ) a, a, α s µ) ). 52) The natural sale for the strong oupling in the short-distane funtion H is s/2. Setting µ = s/2, however, introdues large logarithms of ε in the soft funtion and large logarithms of ν in both the soft and jet funtions. The purpose of this setion is to ontrol these logarithms by the identifiation and solution of renormalization group and evolution equations. The information neessary to perform the resummations is already present in the fatorization 52). The ross setion itself is independent of the fatorization sale µ d dσ ε, ν, s, a) = 0, 53) dµ dεdˆn and of the hoie of the eikonal diretions, ˆξ, used in the fatorization, ln dσ ε, ν, s, a) p J ˆξ ) = 0. 54) dεdˆn 9
In the remainder of this setion we explore the onsequenes of these onditions [27]. 4. Energy flow As a first step, we use the renormalization group equation 53) to organize dependene on the energy flow variable ε. Applying Eq. 53) to the fatorized orrelation 52), we derive the following onsisteny onditions, whih are themselves renormalization group equations: µ d ε s dµ ln S µ, εν, µ d dµ ln J ) s µν ζ ) a, a, α s µ) pj ˆξ ) s µ, µν ζ ) a, a, α s µ) s µ, p J ˆξ ) µ, ˆn, α s µ) µ d dµ ln H = γ s α s µ)), 55) = γ J α s µ)), 56) 2 = γ s α s µ)) + γ J α s µ)). 57) = The anomalous dimensions γ d, d = s, J an depend only on variables held in ommon between at least two of the funtions. Beause eah funtion is infrared safe, while ultraviolet divergenes are present only in virtual diagrams, the anomalous dimensions annot depend on the parameters ν, ε or a. This leaves as arguments of the γ d only the oupling α s µ), whih we exhibit, and ζ, whih we suppress for now. Solving Eqs. 55) and 56) we find S ) ε s s µ, εν, µν ζ ) a, a, α s µ) pj J ˆξ ) s µ, µν ζ ) a, a, α s µ) = S ) ε s s, εν, µ 0 µ 0 ν ζ ) a, a, α s µ 0 ) e = J pj ˆξ µ 0, ) s µ 0 ν ζ ) a, a, α s µ 0 ) e µ µ 0 µ µ 0 dλ λ γsαsλ)), dλ λ γ J αsλ)), 58) 59) for the soft and jet funtions. As suggested above, we will eventually pik µ s to avoid large logs in H. Using these expressions in Eq. 52) we an avoid logarithms of ε or ν in the soft funtion, by evolving from µ 0 = ε s to the fatorization sale µ s. No hoie of µ 0, however, ontrols all logarithms of ν in the jet funtions. Leaving µ 0 free, we find for the ross setion 52) the intermediate result dσ ε, ν, s, a) = dσ 0 s H dε dˆn dˆn µ, p J ˆξ ) µ, ˆn, α s µ) S, εν, ζ ) a, a, α s ε s) ) µ exp dλ ε λ γ s α s λ)) s 20 60)
J pj ˆξ µ 0, ) s µ 0 ν ζ ) a, a, α s µ 0 ) µ exp λ γ J α s λ)). µ 0 dλ We have avoided introduing logarithms of ε into the jet funtions, whih originally only depend on ν, by evolving the soft and the jet funtions independently. The hoie of µ 0 = ε s or s/ν for the soft funtion is to some extent a matter of onveniene, sine the two hoies differ by logarithms of εν. In general, if we hoose µ 0 = s/ν, logarithms of εν will appear multiplied by oeffiients that reflet the size of region Ω. An example is Eq. 4) above. When Ω has a small angular size, µ 0 = s/ν is generally the more natural hoie, sine then logarithms in εν will enter with small weights. In ontrast, when Ω grows to over most angular diretions, as in the study of rapidity gaps [28], it is more natural to hoose µ 0 = ε s. 4.2 Event shape transform The remaining unorganized large logarithms in 60), are in the jet funtions, whih we will resum by using the onsisteny equation 54). The requirement that the ross setion be independent of p J ˆξ implies that the jet, soft and hard funtions obey equations analogous to 55) 57), again in terms of the variables that they hold in ommon [27]. The same results may be derived following the method of Collins and Soper [20], by defining the jets in an axial gauge, and then studying their variations under boosts. For our purposes, only the equation satisfied by the jet funtions [20, 27] is neessary, ln pj p J ˆξ ) ln J ˆξ ) s µ, µν ζ ) a, a, α s µ) ) s = K µ ν ζ ) a, a, α s µ) pj + G ˆξ ) µ, α sµ). 6) The funtions K and G ompensate the ξ -dependene of the soft and hard funtions, respetively, whih determines the kinemati variables upon whih they may depend. In partiular, notie the ombination of ν- and ξ -dependene required by the arguments of the jet funtion, Eq. 45). Sine the definition of our jet funtions 35) is gauge invariant, we an derive the kernels K and G by an expliit omputation of J / ln p J ˆξ ) in any gauge. The multipliative renormalizability of the jet funtion, Eq. 56), with an anomalous dimension that is independent of p J ˆξ ensures that the right-hand side of Eq. 6) is a renormalization-group invariant. Thus, K + G are renormalized additively, and satisfy [20] µ d dµ K ) s µ ν ζ ) a, a, α s µ) pj ˆξ ) µ, α sµ) µ d dµ G 2 = γ K α s µ)), = γ K α s µ)). 62)
Sine G and hene γ K, may be omputed from virtual diagrams, they do not depend on a, and γ K is the universal Sudakov anomalous dimension [20, 29]. With the help of these evolution equations, the terms K and G in Eq. 6) an be reexpressed as [30] ) s K µ ν ζ ) a pj, a, α s µ) + G ˆξ ) µ, α sµ) )) s = K, a, α s ν ζ ) a + G, α s 2 p J ˆξ ) ) 2 2 p J ˆξ s ζ) a /ν dλ λ γ K α s λ )) = B, 2, a, α s 2 p J ˆξ )) 2 p J ˆξ 2 sζ) a /ν dλ λ A, a, α s λ )), 63) where in the seond equality we have shifted the argument of the running oupling in K, and have introdued the notation ) ) B, 2, a, α s µ)) K, a, α s µ) G, α s µ), 2 2A, a, α s µ)) ) γ K α s µ)) + βgµ)) gµ) K, a, α s µ). 64) The primes on the funtions A and B are to distinguish these anomalous dimensions from their somewhat more familiar versions given below. The solution to Eq. 6) with µ = µ 0 is pj J ˆξ ) s s, µ 0 µ 0 ν ζ ) a, a, α s µ 0 ) = J, 2 ζ 0 µ 0 p J ˆξ dλ exp λ B, 2, a, α s 2 λ)) + 2 s/2ζ0 ) ) s µ 0 ν ζ 0) a, a, α s µ 0 ) 2 λ s a/2 ν2 λ) a where we evolve from s/2 ζ 0 ) to p J ˆξ = s/2 ζ ) see Eq. 44)) with dλ A λ, a, α s λ )), 65) ) ν /2 a) ζ 0 =. 66) 2 After ombining Eqs. 59) and 65), the hoie µ 0 = s/2ζ 0 ) = s ζ ν 0) a allows us to ontrol all large logarithms in the jet funtions simultaneously: pj J ˆξ ) s µ, µν ζ ) a, a, α s µ) )) s µ = J,, a, α s exp 2 ζ 0 s/2ζ0 ) dλ λ γ J α s λ)) 22
exp p J ˆξ s/2 ζ0 ) dλ λ B, 2, a, α s 2 λ)) + 2 2 λ s a/2 ν2 λ) a dλ A λ, a, α s λ )). 67) As observed above, we treat a as a fixed parameter, with a small ompared to ln /ε) and ln ν. 4.3 The resummed orrelation Using Eq. 67) in 60), and setting µ = s/2, we find a fully resummed form for the orrelation, dσ ε, ν, s, a) = dσ 0 2 pj H ˆξ )) s, ˆn, α s dε dˆn dˆn s 2 S, εν, ζ ) a, a, α s ε s) ) s/2 exp dλ λ γ s α s λ)) ε s )) 2 s s/2 J,, a, α s exp = 2 ζ 0 dλ λ γ J α s λ)) s/2 ζ0 ) p J ˆξ dλ 2 λ exp λ B dλ, 2, a, α s 2 λ)) + 2 A λ, a, α s λ )). s/2 ζ0 ) s a/2 ν2 λ) a Alternatively, we an ombine all jet-related exponents in Eq. 68) in the orrelation. As we will verify below in Setion 5.2, the ross setion is independent of the hoie of ξ. As a result, we an hoose p J ˆξ s = 2. 69) This hoie allows us to ombine γ J and B in Eq. 68), dσ ε, ν, s, a) = dσ )) 0 s H, ˆn, α s dε dˆn dˆn 2 S, εν,, a, α s ε s) ) s/2 exp dλ )) 2 s ε λ γ s α s λ)) J,, a, α s = 2 ζ 0 s s/2 dλ 2 λ exp λ γ J α s λ)) + B dλ, 2, a, α s 2 λ)) + 2 A λ, a, α s λ )), s/2 ζ0 ) 23 s a/2 ν2 λ) a 68) 70)
with ζ 0 given by Eq. 66). In Eqs. 68) and 70), the energy flow ε appears at the level of one logarithm per loop, in S, in the first exponent. Leading logarithms of ε are therefore resummed by knowledge of γ s ), the one-loop soft anomalous dimension, where we employ the standard notation, ) γ s α s ) = γ s n) αs n 7) n=0 π for any expansion in α s. At the same time, ν appears in up to two logarithms per loop, harateristi of onventional Sudakov resummation. To ontrol ν-dependene at the same level as ε-dependene, it is natural to work to next-to-leading logarithm in ν, by whih we mean the level αs k ln k ν in the exponent. As usual, this requires one loop in B and γ J, and two loops in the Sudakov anomalous dimension A, Eq. 64). These funtions are straightforward to alulate from their definitions given in the previous setions. Only the soft funtion S in Eqs. 68) and 70) ontains information on the geometry of Ω. The exponents are partially proess-dependent, but geometry-independent. In Setion 5, we will derive expliit expressions for these quantities, suitable for resummation to leading logarithm in ε and next-to-leading logarithm in ν. 4.4 The inlusive event shape It is also of interest to onsider the ross setion for e + e -annihilation into two jets without fixing the energy of radiation into Ω, but with the final state radiation into all of phase spae weighted aording to Eq. 4), shematially e + + e J p J, f Ω ) + J 2 p J2, f Ω2 ), 72) where Ω and Ω 2 over the entire sphere. This ross setion an be fatorized and resummed in a ompletely analogous manner. The final state is a onvolution in the ontributions of the jet and soft funtions to ε as in Eq. 26), but with no separate restrition on energy flow into Ω. All partiles ontribute to the event shape. We obtain an expression very analogous to Eq. 68) for this inlusive event shape in transform spae, whih an be written in terms of the same jet funtions as before, and a new funtion S inl for soft radiation as: dσ inl ν, s, a) = dσ 0 2pJ H ˆξ, ˆn, α s ) s/2) dˆn dˆn s S inl 2 = exp )) s ζ ) a, a, α s ν s J,, a, α s 2 ζ 0 p J ˆξ s/2 ζ0 ) dλ λ exp )) exp s/2 dλ s/ν s/2 s/2 ζ0 ) B, 2, a, α s 2 λ)) + 2 24 λ γ s α s λ)) dλ λ γ J α s λ)) 2 λ s a/2 ν2 λ) a dλ A λ, a, α s λ )).